ΛCDM Predictions Research Articles

Measurement of the Weyl potential evolution from the first three years of dark energy survey data.

The Weyl potential, which is the sum of the spatial and temporal distortions of the Universe's geometry, provides a direct way of testing the theory of gravity and the validity of the ΛCDM (Lambda Cold Dark Matter) model. Here we present measurement of the Weyl potential at four redshifts bins using data from the first three years of observations of the Dark Energy Survey. We find that the measured Weyl potential is 2 σ, respectively 2.8 σ, below the ΛCDM predictions in the two lowest redshift bins. We show that these low values of the Weyl potential are at the origin of the tension between Cosmic Microwave Background measurements and weak lensing measurements, regarding the parameter σ8 which quantifies the clustering of matter. Interestingly, we find that the tension remains if no information from the Cosmic Microwave Background is used. Dark Energy Survey data on their own prefer a high value of the primordial fluctuations, together with a slow evolution of the Weyl potential. An important feature of our method is that the measurements of the Weyl potential are model-independent and can therefore be confronted with any theory of gravity, allowing efficient tests of models beyond General Relativity.

High precision accelerator for our hybrid model of the redshift space power spectrum

Abstract Upcoming Large Scale Structure surveys aim to achieve an unprecedented level of precision in measuring galaxy clustering. However, accurately modelling these statistics may require theoretical templates that go beyond two-loop order perturbation theory, especially for achieving precision at smaller scales. In our previous work, we introduced a hybrid model for the redshift space power spectrum of galaxies. This model combines two-loop order templates with N-body simulations to capture the influence of scale-independent parameters on the galaxy power spectrum. However, the impact of scale-dependent parameters was addressed by precomputing a set of input statistics derived from computationally expensive N-body simulations. As a result, exploring the scale-dependent parameter space was not feasible in this approach. To address this challenge, we present an accelerated methodology that utilizes Gaussian processes, a machine learning technique, to emulate these input statistics. Our emulators exhibit remarkable accuracy, achieving reliable results with just 13 N-body simulations for training. Our emulators can reproduce the set of statistics we are interested in with less than 0.1% error in the parameter space within 5σ of the Planck ΛCDM predictions, specifically for scales around k > 0.1 h Mpc−1. Following the training of our emulators, we can predict all inputs for our hybrid model in approximately 0.2 seconds at a specified redshift. Given that performing 13 N-body simulations is a manageable task, our present methodology enables us to construct efficient and highly accurate models of the galaxy power spectra within a manageable time frame.

The circular velocity and halo mass functions of galaxies in the nearby Universe

ABSTRACT The circular velocity function (CVF) of galaxies is a fundamental test of the Lambda cold dark matter ($\Lambda$CDM) paradigm as it traces the variation of galaxy number densities with circular velocity ($v_{\rm {circ}}$), a proxy for dynamical mass. Previous observational studies of the CVF have either been based on H i-rich galaxies, or encompassed low-number statistics and probed narrow ranges in $v_{\rm {circ}}$. We present a benchmark computation of the CVF between $100\,{\text{and}}\,350\ \rm {km\ s^{-1}}$ using a sample of 3527 nearby Universe galaxies, representative for stellar masses between $10^{9.2}\,{\text{and}}\,10^{11.9} \rm {{\rm M}_{\odot }}$. We find significantly larger number densities above 150 $\rm {km\ s^{-1}}$ compared to results from H i surveys, pertaining to the morphological diversity of our sample. Leveraging the fact that circular velocities are tracing the gravitational potential of haloes, we compute the halo mass function (HMF), covering $\sim$1 dex of previously unprobed halo masses ($10^{11.7}{\!-\!}10^{12.7} \rm {{\rm M}_{\odot }}$). The HMF for our sample, representative of the galaxy population with $M_{200}\geqslant 10^{11.35} \rm {{\rm M}_{\odot }}$, shows that spiral morphologies contribute 67 per cent of the matter density in the nearby Universe, while early types account for the rest. We combine our HMF data with literature measurements based on H i kinematics and group/cluster velocity dispersions. We constrain the functional form of the HMF between $10^{10.5}-10^{15.5} \rm {{\rm M}_{\odot }}$, finding a good agreement with $\Lambda$CDM predictions. The halo mass range probed encompasses 72$\substack{+5 -6}$ per cent ($\Omega _{\rm {M,10.5-15.5}} = 0.227 \pm 0.018$) of the matter density in the nearby Universe; 31$\substack{+5 -6}$ per cent is accounted for by haloes below $10^{12.7}\rm {{\rm M}_{\odot }}$ occupied by a single galaxy.

Galaxy dynamics tracing quantum cosmology beyond [formula omitted]CDM below the de Sitter scale of acceleration

It is proposed that the baryonic Tully–Fisher relation (bTFR) and JWST ’Impossible galaxies’ at cosmic dawn are unified in weak gravity by the trace J of the Schouten tensor below the Sitter scale of acceleration adS=cH, where c is the velocity of light and H is the Hubble parameter. Across adS, J parametrizes short-period galaxy rotation curves and fast gravitational collapse beyond the predictions of ΛCDM. The sensitivity of weak gravitation to J=16R, where R is the Ricci scalar tensor, is derived in infrared gravitation from a consistent limit to quantum gravity, reducing to general relativity in the limit of a small Planck constant. For the first time, it identifies the exact relation a0=c2J/2π of the Milgrom parameter across all redshifts, accounting for the bTFR and early galaxy formation accelerated by a factor ∼J1/8. It predicts a0′(0)<0 at the present epoch.

Gravitational imaging through a triple source plane lens: revisiting the ΛCDM-defying dark subhalo in SDSSJ0946+1006

ABSTRACT The ΛCDM paradigm successfully explains the large-scale structure of the Universe, but is less well constrained on subgalactic scales. Gravitational lens modelling has been used to measure the imprints of dark substructures on lensed arcs, testing the small-scale predictions of ΛCDM. However, the methods required for these tests are subject to degeneracies among the lens mass model and the source light profile. We present a case study of the unique compound gravitational lens SDSSJ0946+1006, wherein a dark, massive substructure has been detected, whose reported high concentration would be unlikely in a ΛCDM universe. For the first time, we model the first two background sources in both I- and U-band HST imaging, as well as VLT-MUSE emission line data for the most distant source. We recover a lensing perturber at a 5.9σ confidence level with mass $\log _{10}(M_\mathrm{sub}/{\rm M}_{\odot })=9.2^{+0.4}_{-0.1}$ and concentration $\log _{10}c=2.4^{+0.5}_{-0.3}$. The concentration is more consistent with CDM subhaloes than previously reported, and the mass is compatible with that of a dwarf satellite galaxy whose flux is undetectable in the data at the location of the perturber. A wandering black hole with mass $\log _{10}(M_\mathrm{BH}/{\rm M}_{\odot })=8.9^{+0.2}_{-0.1}$ is a viable alternative model. We systematically investigate alternative assumptions about the complexity of the mass distribution and source reconstruction; in all cases the subhalo is detected at around the ≥5σ level. However, the detection significance can be altered substantially (up to 11.3σ) by alternative choices for the source regularization scheme.

Discord in Concordance Cosmology and Anomalously Massive Early Galaxies

Cosmological parameters are constrained by a wide variety of observations. We examine the concordance diagram for modern measurements of the Hubble constant, the shape parameter from the large-scale structure, the cluster baryon fraction, and the age of the universe, all from non-CMB data. There is good agreement for H0=73.24±0.38kms−1Mpc−1 and Ωm=0.237±0.015. This concordance value is indistinguishable from the WMAP3 cosmology but is not consistent with that of Planck: there is a tension in Ωm as well as H0. These tensions have emerged as progressively higher multipoles have been incorporated into CMB fits. This temporal evolution is suggestive of a systematic effect in the analysis of CMB data at fine angular scales and may be related to the observation of unexpectedly massive galaxies at high redshift. These are overabundant relative to ΛCDM predictions by an order of magnitude at z&gt;7. Such massive objects are anomalous and could cause gravitational lensing of the surface of last scattering in excess of the standard calculation made in CMB fits, potentially skewing the best-fit cosmological parameters and contributing to the Hubble tension.

Modified gravity and massive neutrinos: constraints from the full shape analysis of BOSS galaxies and forecasts for Stage IV surveys

We constrain the growth index γ by performing a full-shape analysis of the power spectrum multipoles measured from the BOSS DR12 data. We adopt a theoretical model based on the Effective Field theory of the Large Scale Structure (EFTofLSS) and focus on two different cosmologies: γCDM and γνCDM, where we also vary the total neutrino mass. We explore different choices for the priors on the primordial amplitude As and spectral index ns , finding that informative priors are necessary to alleviate degeneracies between the parameters and avoid strong projection effects in the posterior distributions. Our tightest constraints are obtained with 3σ Planck priors on As and ns : we obtain γ = 0.647 ± 0.085 for γCDM and γ = 0.612+0.075 -0.090, Mν < 0.30 for γνCDM at 68% c.l., in both cases ∼ 1σ consistent with the ΛCDM prediction γ ≃ 0.55. Additionally, we produce forecasts for a Stage-IV spectroscopic galaxy survey, focusing on a DESI-like sample. We fit synthetic data-vectors for three different galaxy samples generated at three different redshift bins, both individually and jointly. Focusing on the constraining power of the Large Scale Structure alone, we find that forthcoming data can give an improvement of up to ∼ 85% in the measurement of γ with respect to the BOSS dataset when no CMB priors are imposed. On the other hand, we find the neutrino mass constraints to be only marginally better than the current ones, with future data able to put an upper limit of Mν < 0.27 eV. This result can be improved with the inclusion of Planck priors on the primordial parameters, which yield Mν < 0.18 eV.

Distinct distributions of elliptical and disk galaxies across the Local Supercluster as a ΛCDM prediction

Galaxies of different types are not equally distributed in the Local Universe. In particular, the supergalactic plane is prominent among the brightest ellipticals, but inconspicuous among the brightest disk galaxies. This striking difference provides a unique test for our understanding of galaxy and structure formation. Here we use the SIBELIUS DARK constrained simulation to confront the predictions of the standard Lambda Cold Dark Matter (ΛCDM) model and standard galaxy formation theory with these observations. We find that SIBELIUS DARK reproduces the spatial distributions of disks and ellipticals and, in particular, the observed excess of massive ellipticals near the supergalactic equator. We show that this follows directly from the local large-scale structure and from the standard galaxy formation paradigm, wherein disk galaxies evolve mostly in isolation, while giant ellipticals congregate in the massive clusters that define the supergalactic plane. Rather than being anomalous as earlier works have suggested, the distributions of giant ellipticals and disks in the Local Universe and in relation to the supergalactic plane are key predictions of the ΛCDM model.

Testing the cosmological principle with CatWISE quasars: a bayesian analysis of the number-count dipole

ABSTRACT The Cosmological Principle, that the Universe is homogeneous and isotropic on sufficiently large scales, underpins the standard model of cosmology. However, a recent analysis of 1.36 million infrared-selected quasars has identified a significant tension in the amplitude of the number-count dipole compared to that derived from the cosmic microwave background (CMB), thus challenging the Cosmological Principle. Here, we present a Bayesian analysis of the same quasar sample, testing various hypotheses using the Bayesian evidence. We find unambiguous evidence for the presence of a dipole in the distribution of quasars with a direction that is consistent with the dipole identified in the CMB. However, the amplitude of the dipole is found to be 2.7 times larger than that expected from the conventional kinematic explanation of the CMB dipole, with a statistical significance of 5.7σ. To compare these results with theoretical expectations, we sharpen the ΛCDM predictions for the probability distribution of the amplitude, taking into account a number of observational and theoretical systematics. In particular, we show that the presence of the Galactic plane mask causes a considerable loss of dipole signal due to a leakage of power into higher multipoles, exacerbating the discrepancy in the amplitude. By contrast, we show using probabilistic arguments that the source evolution of quasars improves the discrepancy, but only mildly so. These results support the original findings of an anomalously large quasar dipole, independent of the statistical methodology used.

Constraints on the transition redshift from the calibrated gamma-ray burst Ep–Eiso correlation

ABSTRACT We constrain the deceleration–acceleration epoch, namely the transition redshift ztr, adopting model-independent techniques that utilize a calibrated Ep–Eiso correlation for gamma-ray bursts (GRBs). To do so, in addition to real data points, we employ up to 1000 simulated observational Hubble data (OHD) points. We then calibrate the Ep–Eiso correlation by means of the well-consolidate Bézier polynomial technique, interpolating OHD up to the second order. Once GRB data have been calibrated, we consider two strategies of cosmographic expansions, i.e. first we take a direct Hubble rate expansion around ztr, and second the expansion of the deceleration parameter around the same redshift, but with a different order. Employing Type Ia supernovae, baryonic acoustic oscillations and GRB data sets, from Monte Carlo analyses we infer tight constraints on ztr and the jerk parameters at z = ztr, namely jtr. Our results are extremely compatible with previous outcomes and confirm the Lambda cold dark matter predictions, being slightly different in terms of the jerk parameter. In this respect, we conjecture which extensions of the concordance paradigm are possible and we compare our findings with expectations provided by generic dark energy models.

The eROSITA Final Equatorial-Depth Survey (eFEDS) – Splashback radius of X-ray galaxy clusters using galaxies from HSC survey

ABSTRACT We present the splashback radius measurements around the SRG/eROSITA eFEDS X-ray selected galaxy clusters by cross-correlating them with HSC S19A photometric galaxies. The X-ray selection is expected to be less affected by systematics related to projection that affects optical cluster finder algorithms. We use a nearly volume-limited sample of 109 galaxy clusters selected in 0.5–2.0 keV band having luminosity $L_X \gt 10^{43.5}\, {\rm erg \, s}^{-1}\, h^{-2}$ within the redshift z &amp;lt; 0.75 and obtain measurements of the projected cross-correlation with a signal to noise of 17.43. We model our measurements to infer a 3D profile and find that the steepest slope is sharper than −3 and associate the location with the splashback radius. We infer the value of the 3D splashback radius $r_{\rm sp} = 1.45^{+0.30}_{-0.26}\, h^{-1}\, {\rm Mpc}$ . We also measure the weak-lensing signal of the galaxy clusters and obtain halo mass $\log [M_{\rm 200m}/ h^{-1}\, {\rm M_\odot }] = 14.52 \pm 0.06$ using the HSC-S16A shape catalogue data at the median redshift z = 0.46 of our cluster sample. We compare our rsp values with the spherical overdensity boundary $r_{\rm 200m} = 1.75 \pm 0.08\, h^{-1} \, {\rm Mpc}$ based on the halo mass, which is consistent within 1.2σ with the ΛCDM predictions. Our constraints on the splashback radius, although broad, are the best measurements thus far obtained for an X-ray selected galaxy cluster sample.

Mock HI-galaxy catalogs and HI mass functions for future large-scale surveys

One of the interesting modern astrophysics topics is the discrepancy in HI-galaxy number density between the standard model ΛCDM prediction and observations, e.g., ALFALFA and HIPASS. Since then, the simulation over-predicted the abundance and HI mass (MHI) of small satellite and low-mass galaxies. Both problems still require sensitive instruments for wider and deeper HI surveys to increase the number of low-mass HI-galaxy samples. Current HI-galaxy surveys can only reach down to ∼ 107 Solar mass (M⊙). The sensitivity of upcoming radio surveys such as FAST or ASKAP are expected to reach lower MHI and higher redshift(z) HI-galaxy data. Alternatively, the HI source catalogs from cosmological simulations are used to determine the expected HIMF with the upcoming HI survey specification. This work aims to compare the HIMF in various HI conversion recipes. Various HIMF were determined by using the Multidark(Planck) catalog with the box (MB) and the light-cone (ML) types, and were parameterized by the Schechter function. In comparison to the Schechter function, we discovered that the HIMF from cold gas mass (Mcold) conversion method begins to diverge on MHI ≲ 108.5 M⊙. The HIMF from halo mass (Mhalo) recipe, which was constrained by the optical counterpart galaxy clustering, provided a satisfactory relevant shape of HIMF. The best-fit parameter α of MB and ML are equal to 0.00 ± 0.010 and −0.02 ± 0.0026, respectively. The flatter α values from simulations could indicate that the number density of low-MHI is underestimated.

A Population III–Generated Dust Screen at z ∼ 16

The search for alternative cosmological models is largely motivated by the growing discordance between the predictions of ΛCDM and the ever-improving observations, such as the disparity in the value of H 0 measured at low and high redshifts. One model in particular, known as the R h = ct universe, has been highly successful in mitigating or removing all of the inconsistencies. In this picture, however, the anisotropies in the cosmic microwave background (CMB) would have emerged at a redshift z ∼ 16, rather than via fluctuations in the recombination zone at z ∼ 1080. We demonstrate here that a CMB created in the early universe, followed by scattering through a Population III–generated dust screen, is consistent with all of the current data. Indeed, the Planck measurements provide a hint of an ∼2%–4% frequency dependence in the CMB power spectrum, which would be naturally explained as a variation in the optical depth through the dust but not a Thomson scattering–dominated recombination environment. Upcoming measurements should be able to easily distinguish between these two scenarios, e.g., via the detection of recombination lines at z ∼ 1080, which would completely eliminate the dust-reprocessing idea.

Studying large-scale structure probes of modified gravity with COLA

We study the effect of two Modified Gravity (MG) theories, f(R) and nDGP, on three probes of large-scale structure, the real space power spectrum estimator Q 0, bispectrum and voids, and validate fast approximate COLA simulations against full N-body simulations for the prediction of these probes.We find that using the first three even multipoles of the redshift space power spectrum to estimate Q 0 is enough to reproduce the MG boost factors of the real space power spectrum for both halo and galaxy catalogues. By analysing the bispectrum and reduced bispectrum of Dark Matter (DM), we show that the strong MG signal present in the DM bispectrum is mainly due to the enhanced power spectrum. We warn about adopting screening approximations in simulations as this neglects non-linear contributions that can source a significant component of the MG bispectrum signal at the DM level, but we argue that this is not a problem for the bispectrum of galaxies in redshift space where the signal is dominated by the non-linear galaxy bias. Finally, we search for voids in our mock galaxy catalogues using the ZOBOV watershed algorithm.To apply a linear model for Redshift-Space Distortion (RSD) in the void-galaxy cross-correlation function, we first examine the effects of MG on the void profiles entering into the RSD model. We find relevant MG signals in the integrated-density, velocity dispersion and radial velocity profiles in the nDGP theory. Fitting the RSD model for the linear growth rate, we recover the linear theory prediction in an nDGP model, which is larger than the ΛCDM predictionat the 3σ level. In f(R) theory we cannot naively compare the results of the fit with the linear theory prediction as this is scale-dependent, but we obtain results that are consistent with the ΛCDM prediction.

Accelerating Early Massive Galaxy Formation with Primordial Black Holes

Recent observations with JWST have identified several bright galaxy candidates at z ≳ 10, some of which appear unusually massive (up to ∼1011 M ⊙). Such early formation of massive galaxies is difficult to reconcile with standard ΛCDM predictions, demanding a very high star formation efficiency (SFE), possibly even in excess of the cosmic baryon mass budget in collapsed structures. With an idealized analysis based on linear perturbation theory and the Press–Schechter formalism, we show that the observed massive galaxy candidates can be explained with lower SFE than required in ΛCDM if structure formation is accelerated/seeded by massive (≳109 M ⊙) primordial black holes (PBHs) that make a up a small fraction (∼10−6–10−3) of dark matter, considering existing empirical constraints on PBH parameters. We also discuss the potential observational signatures of PBH cosmologies in the JWST era. More work needs to be done to fully evaluate the viability of such PBH models to explain observations of the high-z Universe.

Baryonic solutions and challenges for cosmological models of dwarf galaxies

Galaxies and their dark-matter haloes have posed several challenges to the dark energy plus cold dark matter (ΛCDM) cosmological model. These discrepancies between observations and theory intensify for the lowest-mass (‘dwarf’) galaxies. ΛCDM predictions for the number, spatial distribution and internal structure of low-mass dark-matter haloes have historically been at odds with observed dwarf galaxies, but this is partially expected, because many predictions modelled only the dark-matter component. Any robust ΛCDM prediction must include, hand in hand, a model for galaxy formation to understand how baryonic matter populates and affects dark-matter haloes. In this Review, we consider the most notable challenges to ΛCDM regarding dwarf galaxies, and we discuss how recent cosmological numerical simulations have pinpointed baryonic solutions to these challenges. We identify remaining tensions, including the diversity of the inner dark-matter content, planes of satellites, stellar morphologies and star-formation quenching. Their resolution, or validation as actual problems with ΛCDM, will probably require both refining of galaxy-formation models and improving numerical accuracy in simulations. Galaxies and their dark-matter haloes have posed several challenges to the dark energy plus cold dark matter (ΛCDM) cosmological model. This Review discusses the most notable challenges to ΛCDM regarding dwarf galaxies and the insights provided by recent cosmological numerical simulations.

Cosmological implications of ns ≈ 1 in light of the Hubble tension

Recently, a low-z measurement of the Hubble constant, H0=73.04±1.04km/s/Mpc, was reported by the SH0ES Team. The long-standing Hubble tension, i.e. the difference between the Hubble constant from the local measurements and that inferred from the cosmic microwave background data based on the ΛCDM model, was further strengthened. There are many cosmological models modifying the cosmology after and around the recombination era to alleviate this tension. Some of the models alter the small-scale fluctuation amplitude relative to larger scales, and thus require a significant modification of the primordial density perturbation, especially the scalar spectral index, ns. In certain promising models, ns is favored to be larger than the ΛCDM prediction, and even the scale-invariant one, ns=1, is allowed. In this Letter, we focus on the very early Universe models to study the implication of such unusual ns. In particular, we find that an axiverse with axions in the equilibrium distribution during inflation can be easily consistent with ns=1. This is because the axion behaves as a curvaton with mass much smaller than the inflationary Hubble parameter. We also discuss other explanations of ns different from that obtained based on the ΛCDM.

Evidence for a high-zISW signal from supervoids in the distribution of eBOSS quasars

ABSTRACTThe late-time integrated Sachs-Wolfe (ISW) imprint of $R\gtrsim 100~h^{-1}\, \mathrm{Mpc}$ superstructures is sourced by evolving large-scale potentials due to a dominant dark energy component in the ΛCDM model. The aspect that makes the ISW effect distinctly interesting is the repeated observation of stronger-than-expected imprints from supervoids at z ≲ 0.9. Here we analyse the un-probed key redshift range 0.8 &amp;lt; z &amp;lt; 2.2 where the ISW signal is expected to fade in ΛCDM, due to a weakening dark energy component, and eventually become consistent with zero in the matter dominated epoch. On the contrary, alternative cosmological models, proposed to explain the excess low-z ISW signals, predicted a sign-change in the ISW effect at z ≈ 1.5 due to the possible growth of large-scale potentials that is absent in the standard model. To discriminate, we estimated the high-z ΛCDM ISW signal using the Millennium XXL mock catalogue, and compared it to our measurements from about 800 supervoids identified in the eBOSS DR16 quasar catalogue. At 0.8 &amp;lt; z &amp;lt; 1.2, we found an excess ISW signal with AISW ≈ 3.6 ± 2.1 amplitude. The signal is then consistent with the ΛCDM expectation (AISW = 1) at 1.2 &amp;lt; z &amp;lt; 1.5 where the standard and alternative models predict similar amplitudes. Most interestingly, we also observed an opposite-sign ISW signal at 1.5 &amp;lt; z &amp;lt; 2.2 that is in 2.7σ tension with the ΛCDM prediction. Taken at face value, these recurring hints for ISW anomalies suggest an alternative growth rate of structure in low-density environments at $\sim 100~h^{-1}\, \mathrm{Mpc}$ scales.

A measurement of the scale of homogeneity in the early Universe

We present the first measurement of the homogeneity index, ℋ, a fractal or Hausdorff dimension of the early Universe from the Planck CMB temperature variations δT in the sky. This characterization of the isotropy scale is model-free and purely geometrical,independent of the amplitude of δT.We find evidence of homogeneity (ℋ = 0) for scales larger thanθℋ = 65.9 ± 9.2 deg on the CMB sky. This finding is at odds with the ΛCDM prediction, which assumes a scale invariant infinite universe.Such anomaly is consistent with the well known low quadrupule amplitude in the angular δT spectrum, but quantified in a direct and model independent way. We estimate the significance of our finding for ℋ = 0 using a principal component analysis from the sampling variations of the observed sky. This analysis is validated with theoretical prediction of the covariance matrix and simulations, booth base purely on data or in the ΛCDM prediction.Assuming translation invariance (and flat geometry) we can convert the isotropy scale θℋ into a (comoving) homogeneity scale which is very close to the trapped surface generated by the observed cosmological constant Λ.

The primordial matter power spectrum on sub-galactic scales

ABSTRACTThe primordial matter power spectrum quantifies fluctuations in the distribution of dark matter immediately following inflation. Over cosmic time, overdense regions of the primordial density field grow and collapse into dark matter haloes, whose abundance and density profiles retain memory of the initial conditions. By analysing the image magnifications in 11 strongly lensed and quadruply imaged quasars, we infer the abundance and concentrations of low-mass haloes, and cast the measurement in terms of the amplitude of the primordial matter power spectrum. We anchor the power spectrum on large scales, isolating the effect of small-scale deviations from the Lambda cold dark matter (ΛCDM) prediction. Assuming an analytic model for the power spectrum and accounting for several sources of potential systematic uncertainty, including three different models for the halo mass function, we obtain correlated inferences of $\log _{10}\left(P / P_{\Lambda \rm {CDM}}\right)$, the power spectrum amplitude relative to the predictions of the concordance cosmological model, of $0.0_{-0.4}^{+0.5}$, $0.1_{-0.6}^{+0.7}$, and $0.2_{-0.9}^{+1.0}$ at k = 10, 25, and 50 $\rm {Mpc^{-1}}$ at $68 {{\ \rm per\ cent}}$ confidence, consistent with CDM and single-field slow-roll inflation.